ENDG 319 Section #10 (Hypothesis Testing) Summary Sheet
H0: ;  known and  given. Data are either normally distributed or you are invoking CLT (n  30)
Compute critical value z where
If H1:    0
Reject H0 if z   z
P(Z< -z) = 
Compute statistic
Compute critical value z where
If H1:    0
Reject H0 if z  z
x  0
P(Z> z) = 
z
Compute critical value z/2
 n
Reject H0 if
If H1:    0
where P(Z< -z/2) = /2 and
z   z 2 or z  z 2
P(Z> z/2) = /2
H0: ; ; unknown but n  30 so you invoke CLT. This is a “large sample test for a single mean”;  given.
Compute critical value z where
If H1:    0
Reject H0 if z   z
P(Z< -z) = 
Compute statistic
Compute critical value z where
If H1:    0
Reject H0 if z  z
x  0
P(Z> z) = 
z
Compute critical value z/2
s n
Reject H0 if
If H1:    0
where P(Z< -z/2) = /2 and
z   z 2 or z  z 2
P(Z> z/2) = /2
H0: ;  unknown and data is normally distributed; cannot invoke CLT;  given.
Compute critical value t where
If H1:    0
Reject H0 if t  t
P(T< -t) =  with df = n-1
Compute statistic
Compute critical value t where
If H1:    0
Reject H0 if t  t
x  0
P(T> t) =  with df = n-1
t
Compute critical value t/2
s n
Reject H0 if
If H1:    0
where P(T< -t/2) = /2 and
t  t 2 or t  t 2
P(T> t/2) = /2; df = n-1
H0: d; ,  known and  given. Data are either normally distributed or you invoke CLT (n  30)
Compute critical value z where
If H1: 1  2  d0
Reject H0 if z   z
P(Z< -z) = 
Compute statistic
Compute critical value z where
If H1: 1   2  d 0
Reject H0 if z  z
 x1  x2   d 0
P(Z> z) = 
z
Compute critical value z/2
 12 n1   22 n2
Reject H0 if
If H1: 1   2  d 0
where P(Z< -z/2) = /2 and
z   z 2 or z  z 2
P(Z> z/2) = /2
H0: d; ,  unknown, but n  30: use “large sample test”;  given.
Compute critical value z where
Reject H0 if z   z
If H1: 1  2  d0
P(Z< -z) = 
Compute statistic
Compute critical value z where
If H1: 1   2  d 0
Reject H0 if z  z
x1  x 2   d 0
P(Z> z) = 
z
Compute critical value z/2
s12 n1  s 22 n 2
Reject H0 if
If H1: 1   2  d 0
where P(Z< -z/2) = /2 and
z   z 2 or z  z 2
P(Z> z/2) = /2
H0: d; =  and both are unknown; data are normally distributed;  given.
Compute critical value t where
Compute statistic
Reject H0 if t  t
If H1: 1  2  d0
P(T< -t) =  with df=n1+n2-2

x1  x2   d 0
t
Compute critical value t where
If H1: 1   2  d 0
Reject H0 if t  t
s p 1 n1   1 n2 
P(T> t) =  with df=n1+n2-2
Compute critical value t/2
( n  1)s12  ( n 2  1)s 22
Reject H0 if
with s 2p  1
If H1: 1   2  d 0
where P(T< -t/2) = /2 and
n1  n 2  2
t  t 2 or t  t 2
P(T> t/2) = /2; df=n1+n2-2
Dr. Sameh Nassar

 


 

ENDG 319 Section #10 (Hypothesis Testing) Summary Sheet
H0: d; ≠  and both are unknown; data are normally distributed;  given.
Compute critical value t where
x 1  x 2   d 0 ;
If H1: 1  2  d0
Reject H0 if t  t
t
P(T< -t) = 
2
2
s 1 n1  s 2 n 2
Compute critical value t where
If H1: 1   2  d 0
Reject H0 if t  t
2
P(T> t) = 
s12 n1  s22 n 2
with df 
Compute critical value t/2
2
2
Reject H0 if
s12 n1
s22 n 2
If H1: 1   2  d 0
where P(T< -t/2) = /2 and

t  t 2 or t  t 2
P(T> t/2) = /2
n1 1
n 2 1

 



 


H0: Dd; ≠  and both are unknown; data are normally distributed AND PAIRED;  given.
If H1:  D
 d0
If H1:  D  d 0
If H1:  D
 d0
Compute critical value t where
P(T< -t) = , df = n-1
Compute critical value t where
P(T> t) = , df = n-1
Compute critical value t/2
where P(T< -t/2) = /2 and
P(T> t/2) = /2, df = n-1
Reject H0 if
Compute statistic
t
d  d0
sd n
Reject H0 if
t  t
t  t
Reject H0 if
2 or t  t
t  t
2
Formulating Hypothesis Testing Problems
Hypotheses about a random variable x are often formulated in terms of its distributional
properties. Example, if property is a:
Null hypothesis H0: a = a0
Alternative hypothesis H1: a < a0 || a > a0 || a ≠ a0
Objective of hypothesis testing is to decide whether or not to reject this hypothesis.
Decision is based on estimator â of a:
Reject H0: If observed estimate â lies in rejection region Ra0 (i.e. â  Ra0)
Do not reject H0: Otherwise (i.e. â  Ra0)
Select rejection region to obtain desired error properties:
Type I Error probability α is called the test significance level.
1- is said to be the power of a test; and  is the Type II Error probability
Computing  and choosing sample size
Suppose that we wish to test the hypothesis (H0: ; H1: ) with a significance level 
when 2 is known. For a specific alternative, say =0+, the power of our test is: 1- = P[x>a
when =0+, which is equivalent to:


 

  P Z  z  
  P Z  z 


n 


Dr. Sameh Nassar

z
Hence : n 
 z  2
2

2
or
z
n
 z  2
2
/2
2
( two  tailed)
ENDG 319 Section #10 (Hypothesis Testing) Summary Sheet
H 0:  = 
If H1: 
2
  02
If H1: 
2
  02
If H1: 
2
  02
Compute critical value 21-
where P(2 < 21-) = 
Compute critical value 2
where P(2 > 2) = 
Compute critical values 2/2
and 21-/2
where P(2<21-/2) = /2 and
P(2 > 2/2) = /2
Reject H0 if 2 < 21-
Compute statistic
(n  1)s 2
2 
 02
with df  n  1
Reject H0 if 2 > 2
Reject H0 if
2<21-/2 or 2 > 2/2
H 0:  =  
If H1: ≠ 


If H1:> 
or < 
Compute critical values
F(df1, df2)
where P(F > F) = /2
Compute critical values
F(df1, df2)
where P(F > F) = 
Compute statistic
2
1
2
2
s
if s12  s 22
s
with df1 = n1 - 1 and df 2 = n 2 - 1
or
s2
F  22 if s12  s 22
s1
with df1 = n 2 - 1 and df 2 = n1 - 1
F
(Goodness-for-Fit Test) H0: observation x follows a specified distribution f(x)
Compute statistic
k o  e  2
i
H1: x does not follow the distribution f(x)
2   i
ei
i 1
with df  k  1
Reject H0 if F > F(df1, df2)
Reject H0 if F > F(df1, df2)
Reject H0 if 2 > 2
Note: Expected frequencies ei must be ≥ 5 => collapse classes where necessary
Dr. Sameh Nassar
ENDG 319 Section #11 Summary Sheet
Simple Linear Regression and Correlation